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Dynamical Structure of Irregular Constrained Systems

Olivera Miskovic∗,† and Jorge Zanelli∗

∗Centro de Estudios Cientificos (CECS), Casilla 1469, Valdivia, Chile.†Departamento de Fisica, Universidad de Santiago de Chile,

Casilla 307, Santiago 2, Chile.

Abstract

Hamiltonian systems with functionally dependent constraints (irregular systems), forwhich the standard Dirac procedure is not directly applicable, are discussed. They areclassified according to their behavior in the vicinity of the constraint surface into two fun-damental types. If the irregular constraints are multilinear (type I), then it is possible toregularize the system so that the Hamiltonian and Lagrangian descriptions are equivalent.When the constraints are power of a linear function (type II), regularization is not alwayspossible and the Hamiltonian and Lagrangian descriptions may be dynamically inequivalent.It is shown that the inequivalence between the two formalisms can occur if the kinetic en-ergy is an indefinite quadratic form in the velocities. It is also shown that a system of typeI can evolve in time from a regular configuration into an irregular one, without any catas-trophic changes. Irregularities have important consequences in the linearized approximationto nonlinear theories, as well as for the quantization of such systems. The relevance of theseproblems to Chern-Simons theories in higher dimensions is discussed.

1 Introduction

Dirac’s Hamiltonian analysis provides a systematic method for finding the gauge symmetriesand the physical degrees of freedom of constrained systems like gauge theories and gravity [1].Constraints arise through conditions of the form

φr(z) ≈ 0 (r = 1, ..., R), (1)

where z are local coordinates in phase space Γ. In the most common cases of physical interestthe φ’s are functionally independent; these are the regular constrained systems. There are someexceptional cases in which functional independence is violated. In these irregular systems itis not always clear how to identify symmetries and true degrees of freedom. Moreover, theHamiltonian and Lagrangian descriptions may not be equivalent in irregular systems.

Irregular systems are not necessarily intractable nor exotic. A common example is a relativis-tic massless particle (pµpµ = 0), which is irregular at the origin of momentum space (pµ = 0).There are other physical circumstances in which regularity is violated, and not only for isolatedstates but on large portions of phase space where the system evolves. This is the case in Chern-Simons (CS) theories for dimensions D ≥ 5 where, for some initial configurations, regularity canfail at all times and one is forced to live with this problem.

1

A CS Lagrangian describes a gauge theory for a certain Lie group G in a spacetime of odddimension. The construction is naturally invariant under diffeomorphisms and provides a non-standard but otherwise acceptable description of gravity as a gauge theory [2]-[5]. Furthermore,CS theories are highly nonlinear, possess propagating degrees of freedom [6], and have a veryrich phase space structure with many different sectors, some of which describe irregular systems[7].

In five-dimensional CS supergravity, it was observed that the linearized action around acertain anti-de Sitter background seems to have one more degree of freedom than the fullynonlinear system [8]. This paradoxical behavior can be seen to arise from a violation of theregularity conditions among the symmetry generators of the theory in the region of phase spacedefined by the selected background.

Here we address different scenarios in which regularity conditions can be violated, how suchsystems can be handled in some cases, and why linearization may fail to approximate a nonlinearsystem faithfully. It is found that there are two main types of irregular constraints, multilinear(type I) and nonlinear (type II). The constraints of the first type can always be regularized, whiletype II constraints (of the form fk, where f has a simple zero and k > 1) can be regularizedonly if f is a second class function.

Constraints satisfying regularity conditions are sometimes referred to as effective constraints[9]. The issue of regularity (effectiveness) and its relevance for the equivalence between theLagrangian and Hamiltonian formalisms has also been discussed in [10, 11].

2 Regularity conditions

Consider a dynamical system in a phase space Γ with local coordinates zi ≡ (q, p) (i = 1, . . . , 2n).Conditions (1) define the constraint surface

Σ = z ∈ Γ | φr(z) = 0 (r = 1, . . . , R) (R ≤ 2n) . (2)

Dirac’s procedure guarantees that the system remains on the constraint surface during its evo-lution (for reviews, see Refs. [1],[12]-[16]). Choosing different coordinates on Γ may lead todifferent forms for the constraint functions whose functional independence is not obvious. Theregularity conditions were designed by Dirac to test this [17].

Regularity conditions (RCs): The constraints φr ≈ 0 are regular if and only if theirsmall variations δφr evaluated on Σ define R linearly independent functions of δzi.

To first order in δzi, the variations of the constraints have the form

δφr = Jri δzi (r = 1, ..., R), (3)

where Jri ≡ ∂φr

∂zi

∣

∣

∣

Σis the Jacobian evaluated on the constraint surface. An equivalent definition

of the RCs is [13]: The set of constraints φr ≈ 0 is regular if and only if the Jacobian Jri = ∂φr

∂zi

∣

∣

∣

Σ

has maximal rank, ℜ(J) = R.A simple classical mechanical example of functionally dependent constraints occurs in a 2-

dimensional phase space with coordinates (q, p) and constraints φ1 ≡ q ≈ 0 and φ2 ≡ pq ≈ 0.

In this case, J =

[

1 p0 q

]

q=0

and ℜ (J) = 1.

2

A system of just one constraint can also fail the test of regularity. Consider for example theconstraint φ = q2 ≈ 0 in a 2-dimensional phase space. In this case, J = (2q, 0)q2=0

= 0 and

ℜ (J) = 0. The same problem occurs with the constraint qk ≈ 0, for k > 1, which has a zeroof k-th order on the constraint surface. This example illustrates that one constraint may bedependent on itself, while one function is, by definition, always functionally independent.

Equivalence: Different sets of constraints are said to be equivalent if they define the sameconstraint surface.

Note that this definition refers to the locus of constraints, not to equivalence of the resultingdynamics. Since the surface Σ is defined by the zeros of the constraints, while the regularityconditions depend on their derivatives, it is possible to replace a set of irregular φ’s by anequivalent set of regular constraints φ.

In the classification of irregular systems, two questions present themselves: what is the natureof the constraints that give rise to irregularity, and where the irregularities can occur. Theseissues are addressed in the following subsections. A third question is whether a system canevolve from an initial state in which regularity holds, into an irregular configuration. This willbe discussed in the last section.

2.1 Basic types of irregular constraints

Irregular constraints can be classified according to their behavior in the vicinity of the surfaceΣ. For example, linearly dependent constraints have Jacobian with constant rank R′ throughoutΣ, and

φr ≡ Jri (z)

(

zi − zi)

≈ 0, ℜ(J) = R′ < R. (4)

These constraints are regular systems in disguise simply because R−R′ constraints are redundantand should be discarded. The subset with R′ linearly independent constraints gives the correctdescription. For example, the linearly dependent constraints φ1 = z and φ2 = 2z are clearly inthis category. Apart from this trivial case, two main types of truly irregular constraints, whichdo not possess a linear approximation in the vicinity of some points of Σ, can be distinguished:

Type I. Multilinear constraints. Consider the constraint

φ ≡

M∏

i=1

fi(z) ≈ 0, (5)

where the functions fi have simple zeros. Each factor defines a surface of codimension 1,

Σi ≡ z ∈ Γ | fi(z) = 0 , (6)

and Σ is the collection of all surfaces, Σ =⋃

Σi. The rank of Jacobian of φ is reduced atintersections

Σij ≡ Σi

⋂

Σj. (7)

Thus, the RCs hold everywhere on Σ, except at the intersections Σij, where φ has zeros of higherorder. Note that the intersections (7) also include the points where more than two Σ’s overlap.

3

Type II. Nonlinear constraints. Consider the constraint of the form

φ ≡ [f(z)]k ≈ 0 (k > 1) , (8)

where the function f(z) has a simple zero. This constraint has a zero of order k in the vicinity ofΣ, its Jacobian vanishes on the constraint surface and, therefore, the RCs fail (here we assumek > 1 in order to avoid infinite values for ∂φ

∂zi on Σ). It could seem harmless to replace φ by theequivalent regular constraint f(z) ≈ 0, but it turns out that this may change the dynamics oforiginal system, as we show below.

Types I and II are the two fundamental generic classes of irregular constraints. In general,there can be combinations of them occurring simultaneously near a constraint surface, as inconstraints of the form φ = [f1(z)]2 f2(z) ≈ 0, etc.

2.2 Classification of constraint surfaces

The previous classification refers to the way in which φ approaches zero. Now we will discusswhere regularity can be violated. The rank of the Jacobian ∂φr

∂zi need not be constant throughoutΣ : suppose one eigenvalue of the Jacobian vanishes on a submanifold Σ0 ⊂ Σ. On Σ0 regularityis violated, while it still holds on the rest of Σ. Thus, barring accidental degeneracies such aslinearly dependent constraints, one of these three situation may present themselves:

A. The RCs are satisfied everywhere on the constraint surface: J has maximal rank through-out Σ (regular systems).

B. The RCs fail everywhere on the constraint surface: J has constant rank R′ < R on Σ.C. The RCs fail on Σ0: ℜ (J)|

Σ0= R′ < R, while ℜ (J) = R elsewhere on Σ.

In the last case, the constraint surface can be decomposed into two non overlapping sets Σ0

and ΣR. Then, the rank of the Jacobian jumps from ℜ (J) = R on ΣR, to ℜ (J) = R′ on Σ0.Although the functions φr are continuous and differentiable, this is not sufficient for regularity.Irregular cases are illustrated by the following examples.

In a (2 + N)-dimensional phase space (q, p, z1, . . . , zN ), the constraints φ1 ≡ q − p ≈ 0and φ2 ≡ qp ≈ 0 are irregular on the whole constraint surface

(0, 0, z1, . . . , zN )

, where theJacobian has rank ℜ(J) = 1. Note that these constraints are always irregular, although thefunctions q +p and qp are functionally independent everywhere except for q = p, which happensto be the case at the constraint surface.

An example having both regular and irregular sectors is a massless relativistic particle inMinkowski space with phase space (qµ, pν). The constraint φ ≡ pµpµ ≈ 0 has Jacobian J =(0, 2pµ)φ=0, and its rank is one everywhere, except at the apex of the cone, pµ = 0, where thelight-cone is not differentiable and the Jacobian has rank zero.

The lack of regularity, however, is not necessarily due to the absence of a well defined smoothtangent space for Σ. Consider for example the multilinear constraint

φ(x, y, z) = (x − 1)(x2 + y2 − 1) ≈ 0. (9)

Here the constraint surface Σ is composed of two sub-manifolds: the plane Π = (x, y, z) |x − 1 ≈ 0, and the cylinder C = (x, y, z) | x2 + y2 − 1 ≈ 0, which are tangent to each otheralong the line L = (x, y, z) | x = 1, y = 0, z ∈ R. The Jacobian on Σ is

J =(

3x2 + y2 − 2x − 1, 2y (x − 1) , 0)

φ=0(10)

4

and its rank is 1 everywhere, except on L, where it is zero. The constraint φ is irregular on thisline. However, the tangent vectors to Σ are well defined there. The irregularity arises becauseφ is a multilinear constraint of the type described by (5) and has two simple zeros overlappingon L. The equivalent set of regular constraints on L is φΠ = x− 1 ≈ 0, φC = x2 + y2 − 1 ≈ 0,as we will see below.

3 Treatment of irregular systems

In what follows regular systems and linearly dependent constraints will not be discussed. Theyare either treated in standard texts, or they can be trivially reduced to the regular case.

3.1 Multilinear constraints

Consider a system of type I, as in Eq. (5). In the vicinity of an irregular point where only twosurfaces (6) intersect, say Σ1 and Σ2, the constraint φ ≈ 0 is equivalently described by the setof regular constraints

f1 ≈ 0, f2 ≈ 0. (11)

This replacement generically changes the Lagrangian of the system, and the orbits, as well.Suppose the original canonical Lagrangian is

L(q, u) = piqi − H(q, p) − uφ(q, p), (12)

where H is the Hamiltonian containing all regular constraints. Replacing φ by (11), gives riseto an effective Lagrangian

L12(q, v) = piqi − H(q, p) − v1f1(q, p) − v2f2(q, p). (13)

defined on Σ12. Thus, instead of the irregular Lagrangian (12) defined on the whole Σ, thereis a collection of regularized effective Lagrangians defined in the neighborhood of the differentintersections of Σis. For each of these regularized Lagrangians, the Dirac procedure can becarried out to the end.

Let us illustrate this with the example of a Lagrangian in a (2 + N)-dimensional configurationspace (x, y, q1, . . . , qN ),

L =1

2

N∑

k=1

(

qk)2

+1

2

(

x2 + y2)

− λxy. (14)

This Lagrangian describes a free particle moving on the set(

x, y, qk)

∈ RN+2

∣

∣

∣xy = 0

⊂ RN+2, (15)

which is the union of two (N + 1)-dimensional planes where x and y vanish respectively. Theconstraint surface defined by xy = 0 can be divided into the following sets:

Σ1 =(

x, 0, qk; px, py, pk

)∣

∣

∣x 6= 0

Σ2 =(

0, y, qk; px, py, pk

)∣

∣

∣y 6= 0

(16)

Σ12 =(

0, 0, qk; px, py, pk

)

.

5

The constraint is regular on Σ1

⋃

Σ2, while on Σ12 it is irregular and can be exchanged byφ1 = x ≈ 0, φ2 = y ≈ 0. The corresponding regularized Lagrangians are

L1 =1

2

N∑

k=1

(

qk)2

+1

2x2,

L2 =1

2

N∑

k=1

(

qk)2

+1

2y2 (17)

L12 =1

2

N∑

k=1

(

qk)2

,

and the Lagrange multipliers have dropped out, so the regularized Lagrangians describe physicaldegrees of freedom only – as expected.

The corresponding regularized Hamiltonians are

H1 =1

2

N∑

k=1

p2k +

1

2p2

x,

H2 =1

2

N∑

k=1

p2k +

1

2p2

y (18)

H12 =1

2

N∑

k=1

p2k,

which are defined in the corresponding reduced manifolds of phase space (obtained after com-pleting the Dirac procedure):

Σ1 =(

x, 0, qk; px, 0, pk

)∣

∣

∣x 6= 0

Σ2 =(

0, y, qk; 0, py, pk

)∣

∣

∣y 6= 0

(19)

Σ12 =(

0, 0, qk ; 0, 0, pk

)

.

It is straightforward to generalize the proposed treatment when more than two surfaces Σi

overlap.

Evolution of a multilinearly constrained system. Since in the presence of a multi-linear constraint there are regions of the phase space where the Jacobian has different rank,a question arises about the evolution of this system. Can the system evolve from a genericconfiguration in a region of maximal rank, reaching a configuration of lower rank in finite time?In the case that that were possible, what happens with the system afterwards?. (This problemshould not be confused with the issues arising in degenerate systems [18]-[20].)

To answer this question let us consider the simple example discussed above (14), for N = 1,

L =1

2

(

x2 + y2 + z2)

− λxy. (20)

6

In the regions Σ1 and Σ2 [see Eqs. (16)], the rank is maximal and the free particle can movefreely along the x- or y-axis respectively.

Suppose that the initial state is

x(0) = a > 0, y(0) = 0, z(0) = 0, x(0) = −v < 0, y(0) = 0, z(0) = 0, (21)

so that the particle is moving on Σ1, with finite speed along the x-axis towards x = 0 on Σ12.The evolution is given by x(t) = a− vt, y(t) = 0, z(t) = 0 and the particle clearly reaches x = 0in a finite time (T = a/v). What happens then? According to the evolution equation, for x < 0the trajectory takes the form x(t) = a′ − v′t, y(t) = 0, z(t) = 0, however the action would beinfinite unless a = a′ and v = v′. Therefore, the particle continues unperturbed past beyond thepoint where the RCs fail. So, the irregular surface is not only reachable in a finite time, but itis crossed without any observable effect on the trajectory.

¿From the point of view of the trajectory in phase space, it is clear that the initial state(a, 0, 0;−v, 0, 0) lies on the surface Σ1, and at t = T the system reaches the point (0, 0, 0;−v, 0, 0),which does not lie on the surface Σ12 = (0, 0, z; 0, 0, pz ).

While it is true that at t = T the Jacobian changes rank, it would be incorrect to concludethat the evolution suffers a jump since the dynamical equations are perfectly valid there. Inorder to have significant change in dynamics, the Jacobian should change its rank in an openset.

3.2 Nonlinear constraints

Let us now turn to the case of irregular systems of type II. As we will show, it is possibleto replace a nonlinear irregular constraint by an equivalent linear one without changing thedynamical contents of the theory, provided the linear constraint is second class. Otherwise,the resulting Hamiltonian dynamics will be, in general, inequivalent to that of the originalLagrangian system.

In order to illustrate this point, consider a system given by the Lagrangian

L(q, u) =1

2γij qiqj − u [f(q)]k , (22)

where k > 1 and

f(q) ≡ ciqi 6= 0, i = 1, ..., N. (23)

Here we assume the metric γij to be constant and invertible, and the coefficients ci are alsoconstant. The Euler-Lagrange equations describe a free particle in an N -dimensional space, withtime evolution qi(t) = vi

0 t+ qi0, where u(t) is a Lagrange multiplier. This solution is determined

by 2N initial conditions, qi(0) = qi0 and qi(0) = vi

0 subject to the constraints ciqi0 = 0 and

civi0 = 0. Thus, the system possesses N − 1 physical degrees of freedom.In the Hamiltonian approach this system has a primary constraint π ≡ ∂L2

∂u≈ 0 whose

preservation in time leads to the secondary constraint

φ ≡ [f(q)]k ≈ 0. (24)

7

According to (8), this is a nonlinear constraint and there are no further constraints. As aconsequence, the system has only two first class constraints

π ≈ 0, fk ≈ 0

, and N −1 degreesof freedom, as found in the Lagrangian approach.

On the other hand, if one chooses instead of (24), the equivalent linear constraint

f(q) = ciqi ≈ 0, (25)

then its time evolution yields a new constraint,

χ(p) ≡ γijci pj ≈ 0. (26)

Now, sincef, χ = γijci cj ≡ ‖c‖2 , (27)

two cases can be distinguished:

• If ‖c‖ = 0, there are three first class constraints, π ≈ 0, f ≈ 0 and χ ≈ 0, which meansthat the system has N − 2 physical degrees of freedom. In this case, substitution of (24)by the equivalent linear constraint (25), yields a dynamically inequivalent system.

• If ‖c‖ 6= 0, then f ≈ 0 and χ ≈ 0 are second class, while π ≈ 0 is first class, which leavesN − 1 physical degrees of freedom and the substitution does not change the dynamics ofthe system.

Thus, if fk ≈ 0 is irregular, replacing it by the regular constraint f ≈ 0 changes the dynamicsif f is a first class function, but it gives the correct result if it is a second class function.

Note that in the Lagrangian description there is no room to distinguish first and second classconstraints, so it would seem like the value of ||c|| didn’t matter. However, the inequivalenceof the substitution can be understood in the Lagrangian analysis as well. Suppose that it werepermissible to exchange the constraint fk ≈ 0 by f ≈ 0 in the Lagrangian. Then, instead of(22), one would have

L(q, u) =1

2γij qiqj − uf(q). (28)

It can be easily checked that (28) yields N − 2 degrees of freedom when ‖c‖ = 0, and N − 1degrees of freedom when ‖c‖ 6= 0, which agrees with the results obtained in the Hamiltoniananalysis. Note that the substitution of fk by f modifies the dynamics only if γijci cj = 0, butthis can happen nontrivially only if the metric γij is not positive definite.

In general, a nonlinear irregular constraint φ ≈ 0 has a multiple zero on the constraintsurface Σ, which means that its gradient vanishes on Σ as well. An immediate consequence of(

∂φ/∂zi)

≈ 0, is that φ commutes with all finite functions on Γ,

φ, F (z) ≈ 0. (29)

As a consequence, φ ≈ 0 is first class and is always preserved in time,

φ ≈ 0. (30)

8

On the other hand, a nonlinear constraint cannot be viewed as a symmetry generator simplybecause it does not generate any transformation,

δεzi =

zi, εφ

≈ 0. (31)

Consistently with this, φ cannot be gauge-fixed, as there is no finite function P on Γ such that

φ,P 6= 0. (32)

In this sense, a nonlinear first class constraint that cannot be gauge-fixed, cancels only half adegree of freedom. The other half degree of freedom cannot be cancelled because the gauge-fixingfunction does not exist and, in particular, it cannot appear in the Hamiltonian. Although thefeatures (29-32) allow counting the degrees of freedom in a theory, these systems are pathologicaland their physical relevance is questionable since their Lagrangians cannot be regularized.

When a nonlinear constraint φ ≈ 0 can be exchanged by a regular one, the Lagrangian isregularized as in the case of multilinear constraints. For example, the system (22) with ‖c‖ 6= 0has Hamiltonian

H =1

2γijpipj + λπ + uf(q), (33)

where f = ciqi will turn out to be a second class constraint. The corresponding regularized

Lagrangian is

Lreg =1

2γij qiqj − uf(q), (34)

which coincides with L, Eq. (28), as expected.In the Refs. [10, 11] irregular systems of the type II were discussed. It was pointed out that

there was a possible loss of dynamical information in some cases. From our point of view, it isclear that this would occur when f is a first class function.

4 Linearization of irregular systems

It has been observed in five dimensional Chern Simons theory, that the effective action for thelinearized perturbations of the system around certain backgrounds seems to have more degreesof freedom than the fully nonlinear theory [8]. This is puzzling since the heuristic picture isthat the degrees of freedom of a system correspond to the small perturbations around a localminimum of the action, and therefore the number of degrees of freedom should not change whenthe linearized approximation is used.

In view of the discussion in the previous section, it is clear that a possible solution of thepuzzle lies in the fact that substituting a nonlinear constraint by a linear ones may change thedynamical features of the theory. But the problem with linear approximations is more serious:the linearized approximation retains only up to quadratic and bilinear terms in the Lagrangian,which give linear equations for the perturbations. Thus, irregular constraints in the vicinity ofthe constraint surface are erased in the linearized action. The smaller number of constraintsin the effective theory can lead to the wrong conclusion that the effective system possess moredegrees of freedom than the unperturbed theory. The lesson to be learned is that the linearapproximation is not valid in the part of the phase space where the RCs fail.

9

This is illustrated by the same example discussed earlier (22). One can choose as a back-ground the solution (q1, . . . , qN , u), where qi(t) = qi

0 + vi0t satisfies the constraint

ciqi = 0, (35)

and u(t) is an arbitrarily given function. This describes a free particle moving in the (N − 1)-dimensional plane defined by (35). The linearized effective Lagrangian, to second order in thesmall perturbations si = qi − qi(t) and w = u − u(t), has the form

Leff (s,w) =1

2γij

(

vi0 + si

)

(

vj0 + sj

)

− u(

cisi)2

, (36)

and the equations of motion are

si + Γij(t) sj = 0 i = 1, . . . , N, (37)

where Γij ≡ 2u γikckcj is the eigen frequency matrix. Since u is not a dynamical variable, it is

not varied and the nonlinear constraint(

cisi)2

= 0 is absent from the linearized equations. Thesystem described by (37) possesses N physical degrees of freedom, that is, one degree of freedommore than the original nonlinear theory (22).

The only indication that one of these degrees of freedom has a nonphysical origin is thefollowing: If ‖c‖ 6= 0, splitting the components of si along ci and orthogonal to ci as

si(t) ≡ s(t)γijcj + si⊥(t), (38)

gives rise to the projected equations

si⊥ = 0 (39)

s + 2u(t)‖c‖2s = 0. (40)

The N − 1 components of si⊥(t) obey a deterministic second order equation, whereas s(t)

satisfies an equation which depends on an indeterminate arbitrary function u(t). The dependenceof s = s(t, u(t)) on the background Lagrange multiplier u is an indication that s is a nonphysicaldegree of freedom, since u was an arbitrary function to begin with. This is not manifest in Eq.(40), where u is a fixed function and, from a naive point of view, s(t) is determined by the sameequation, regardless of the physically obscure origin of the function u. It is this naive analysisthat leads to the wrong conclusion indicated above.

Let us emphasize that a linearized theory may be consistent by itself, but it is not necessarilya faithful approximation of a nonlinear theory.

One way to avoid the inconsistencies between the original theory and the linearized onewould be to first regularize the constraints (if possible) and then linearize the correspondingregular Lagrangian.

5 Chern-Simons theories

Hamiltonian structure of CS theories has been studied in [6]. The phase space of a CS theoryin D = 2n + 1 space-time dimensions, invariant under N -parameter gauge group, is defined

10

by canonically conjugate pairs of fields (Aai (x, t), πi

a(x′, t)), where a = 1, . . . , N and xi (i =

1, . . . , 2n) are the local coordinates on a spatial section. The CS Hamiltonian density is givenby

H = Aa0 Ga + ua

i φia, (41)

where uai (x, t) and Aa

0(x, t) are Lagrange multipliers for the constraints

φia ≡ πi

a − Lia(A

bj) ≈ 0, (42)

Ga = gaa1···anF a1 ∧ · · · ∧ F an ≈ 0. (43)

Here Ga ≡ d2nx Ga, g is a symmetric tensor of rank n + 1, invariant under action of a gaugegroup, and F = dA + A ∧ A is the curvature 2-form associated to the gauge field 1-form A.

Constraints φia are regular because they are linear in momenta. Thus, the regularity of

CS theories is determined by momentum-independent constraints Ga. Their small variations,δGa = Jab DδAb, evaluated at Ga = 0, give the (2n− 2)-form Jab, which can be identified as theJacobian,

Jab ≡ n gaba2···anF a2 ∧ · · · ∧ F an . (44)

According to Dirac’s definition, sufficient and necessary condition for Ga to be regular is

ℜ (Jab) = N. (45)

Since Jab is field dependent, its rank may change in space. In particular, for a pure gaugeconfiguration F = 0, and Jab has rank zero. For other configurations, the rank of Jacobiancan range from zero to N , and the irregularities are always of multilinear type because in theexpression (43) the phase space coordinates Aa

i occur only linearly.In general, a non-abelian CS theory for D ≥ 5 possesses a non-vanishing number of physical

degrees of freedom [6]f2n+1(N) = nN − n − N (N ≥ 2) (46)

in regular and generic cases [20]. Therefore, in CS theories, the study of dynamics requires notonly the analysis of regularity, but also of genericity. In spite of the fact that both conditionsare expressed in terms of the same matrix Ωij

ab, they are independent. For example, the extremecase of F = 0 is both irregular and degenerate, but there are examples in CS supergravity whichare generic and irregular [8]. The opposite case occurs in a five-dimensional CS theory basedon G1 × G2 for particular choice of invariant tensor. In this case, there exist configurationswhich are regular but degenerate. Take the group indices as a = (r, α) corresponding to G1

and G2 respectively, and invariant tensor as grs1 = grs and gαβ1 = gαβ (both invertible). Then

the configuration F a =(

f1dx1 ∧ dx2, h1dx3 ∧ dx4

)

is regular and degenerate. Indeed, Jab =(

grs f1 0

0 gαβ h1

)

is regular, while Ωijab with non-vanishing components Ω34

rs = grs f1 and Ω12αβ =

gαβ h1 has 2N zero modes and is therefore degenerate.As a consequence of existence of both regularity and genericity issues, the regularization

problem is much more delicate in CS theories.

11

6 Comments

1 - Dirac conjecture. Dirac conjectured that all first class constraints generate gaugesymmetries [17]. It was shown that Dirac’s conjecture is not true for first class constraints ofthe form fk (k > 1), and following from f ≈ 0 [14, 15]. Therefore, for systems with nonlinearconstraints, the conjecture does not work and they generically provide counterexamples of it[13, 21, 22].

From the point of view of irregular systems, it is clear that Dirac’s conjecture fails fornonlinear constraints because they do not generate any local transformation, c.f. Eq. (31). InRefs. [10, 11] it was observed that Dirac’s conjecture may not hold in the presence of irregularconstraints of type II.

In the case of multilinear constraints, however, Dirac’s conjecture holds. The fact that atirregular points the constraints do not generate any transformation only means that these arefixed points of the gauge transformation.

2 - Quantization. Although, in view of the above discussion, it is possible to deal system-atically with classical theories containing irregular constraints, there may be severe problems intheir quantum description. Consider a path integral of the form

Z ∼

∫

[dq][dp][du] exp i [pq − H(q, p) − uφ(q, p)] , (47)

where φ = [f(q, p)]k is a nonlinear constraint. Integration on u yields to δ(

fk)

, which is notwell-defined for a zero of order k > 1, making the quantum theory ill defined. Only if thenonlinear constraint could be exchanged by the regular one, f(q, p) ≈ 0, the quantum theorycould be saved. An example of this occurs in the standard approach for QED, where it isusual practice to introduce the nonlinear gauge fixing term u(∂iA

i)2 in order to fix the primary

constraint π0 ≡(

δIED/δA0

)

≈ 0. Since the gauge condition f(A) = ∂iAi(x) ≈ 0 is a second

class constraint, its substitution by a regular constraint does not change its dynamical structure.

7 Summary

We have discussed the dynamics and evolution of a system possessing constraints which mayviolate the regularity conditions (functional independence) on some subsets of the constraintsurface Σ. These so-called irregular systems are seen to arise generically because of nonlinearitiesin the constraints and can be classified into two families: multilinear (type I) and nonlinear (typeII).

• Type I constraints are of the form φ =∏

fi(z), where fi possess simple zeros. Theseconstraints violate the regularity conditions (RCs) on sets of measure zero on the constraintsurface Σ.

• Type I constraints can be exchanged by equivalent constraints which are regular giving anequivalent dynamical system.

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• Type II constraints are of the form φ = fk (k > 1) where f has a simple zero. They violatethe RCs on sets of nonzero measure on Σ.

• A type II constraint can be replaced by an equivalent linear one only if the latter issecond class; if the equivalent linear constraint is first class, substituting it for the originalconstraint would change the system.

• In general, the orbits can cross the configurations where the RCs are violated withoutany catastrophic effect for the system. If the symplectic form degenerates at the irregularpoints, additional analysis is required.

• The naive linearized approximation of an irregular constrained system generically changesit by erasing the irregular constraints. In order to study the perturbations around aclassical orbit in an irregular system, it would be necessary to first regularize it (if possible)and only then do the linearized approximation.

• Chern-Simons theories possess irregular and regular sectors. This problem is independentof the presence of degeneracies in the symplectic form, making the regularization problemmuch more complex.

8 Acknowledgments

We are grateful to Milutin Blagojevic, Marc Henneaux and Claudio Teitelboim for enlighteningand helpful comments, and specially to Ricardo Troncoso for many useful discussions, insightsand friendly criticisms. We would also like to thank Josep Maria Pons and Antonio Garcia-Zenteno for bringing their work to our attention and for indicating its relevance to ours. Thiswork is partially funded by grants FONDECYT 1020629, 7020629, 1010450, 2010017 and thegrant MECESUP USA 9930. One of us (O. M.) thanks the Abdus Salam ICTP for hospitalityduring the completion of this work. The generous support of Empresas CMPC to CECS is alsoacknowledged. CECS is a Millennium Science Institute and is funded in part by grants fromFundacion Andes and the Tinker Foundation.

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aj = F a

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ab

is largest, ℜ(

Ωijab

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= 2nN − 2n.

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